# What is the probability that among the 12 months of the year there are 3 non necessarily consecutive months containing exactly 4 birthdays?

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Given 20 people, what is the probability that among the 12 months of the year there are 3 non necessarily consecutive months containing exactly 4 birthdays?

hints:

1. to count the number of elements of the state space, look at the following proposition:

There are ( n + r -1 choose r-1 ) distinct nonnegative integer valued vectors

(x_1, x_2, ... , x_3) satisfying

x_1 + x_2 + ... x_r = n

hint 2:

assume that each month has the same number of days, so that the probability that a birthday falls in a particular month is 1/12.

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This solution is comprised of a detailed explanation to answer what is the probability that among the 12 months of the year there are 3 non necessarily consecutive months containing exactly 4 birthdays.

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Let C(n,m) denote the number to choose m from n.

1. From hint1, we put 20 people into 12 months, there are x_1 people in Jan, x_2 people in Feb, ..., x_12 people in Dec and x_1 + x_2 + ... + ...

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